Signals and Systems/Aperiodic Signals

Aperiodic Signals

The opposite of a periodic signal is an aperiodic signal. An aperiodic function never repeats, although technically an aperiodic function can be considered like a periodic function with an infinite period.

Background

If we consider aperiodic signals, it turns out that we can generalize the Fourier Series sum into an integral named the Fourier Transform. The Fourier Transform is used similarly to the Fourier Series, in that it converts a time-domain function into a frequency domain representation. However, there are a number of differences:

1. Fourier Transform can work on Aperiodic Signals.
2. Fourier Transform is an infinite sum of infinitesimal sinusoids.
3. Fourier Transform has an inverse transform, that allows for conversion from the frequency domain back to the time domain.

Fourier Transform

The Fourier Transform is the following integral:

$ℱ(f(t))=F(j\omega )=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t)e^{-j\omega t}dt$

Examples of Fourier Transforms:

1. Unit impulse function, $\delta (t)$: $ℱ(\delta (t))=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (t)e^{-j\omega t}dt=1$
2. Rectangular pulse function
3. $\exp (-at)u(t)a>0$

Inverse Fourier Transform

And the inverse transform is given by a similar integral:

${ℱ}^{-1}\{F(j\omega )\}=f(t)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(j\omega )e^{j\omega t}d\omega$

Using these formulas, time-domain signals can be converted to and from the frequency domain, as needed.

Partial Fraction Expansion

One of the most important tools when attempting to find the inverse fourier transform is the Theory of Partial Fractions. The theory of partial fractions allows a complicated fractional value to be decomposed into a sum of small, simple fractions. This technique is highly important when dealing with other transforms as well, such as the Laplace transform and the Z-Transform.

Duality

The Fourier Transform has a number of special properties, but perhaps the most important is the property of duality.

We will use a "double-arrow" signal to denote duality. If we have an even signal f, and it's fourier transform F, we can show duality as such:

$f(t)\iff F(j\omega )$

This means that the following rules hold true:

$ℱ\{f(t)\}=F(j\omega )$ AND $ℱ\{F(t)\}=f(j\omega )$

Notice how in the second part we are taking the transform of the transformed equation, except that we are starting in the time domain. We then convert to the original time-domain representation, except using the frequency variable. There are a number of results of the Duality Theorem.

Convolution Theorem

The Convolution Theorem is an important result of the duality property. The convolution theorem states the following:

Or, another way to write it (using our new notation) is such:

$A\times B\iff A*B$

Signal Width

Another principle that must be kept in mind is that signal-widths in the time domain, and bandwidth in the frequency domain are related. This can be summed up in a single statement:

Thin signals in the time domain occupy a wide bandwidth. Wide signals in the time domain occupy a thin bandwidth.

This conclusion is important because in modern communication systems, the goal is to have thinner (and therefore more frequent) pulses for increased data rates, however the consequence is that a large amount of bandwidth is required to transmit all these fast, little pulses.

Power and Energy

Energy Spectral Density

Unlike the Fourier Series, the Fourier Transform does not provide us with a number of discrete harmonics that we can add and subtract in a discrete manner. If our channel bandwidth is limited, in the Fourier Series representation, we can simply remove some harmonics from our calculations. However, in a continuous spectrum, we do not have individual harmonics to manipulate, but we must instead examine the entire continuous signal.

The Energy Spectral Density (ESD) of a given signal is the square of its Fourier transform. By definition, the ESD of a function f(t) is given by F²(jω). The power over a given range (a limited bandwidth) is the integration under the ESD graph, between the cut-off points. The ESD is often written using the variable E_f(jω).

$E_f(j\omega )=F^2(j\omega )$

Power Spectral Density

The Power Spectral Density (PSD) is similar to the ESD. It shows the distribution of power in the spectrum of a particular signal.

$P_f(j\omega )=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(j\omega )d\omega$

Power spectral density and the autocorrelation form a Fourier Transform duality pair. This means that:

$P_f(j\omega )=ℱ[R_{ff}(t)]$

If we know the auto correlation of the signal, we can find the PSD by taking the Fourier transform. Similarly, if we know the PSD, we can take the inverse Fourier transform to find the auto correlation signal.